Whether vertical integration between a downstrean oligopolist and an upstream oligopolist is profitable for an integrated pair of firms is shown to depend on how one formulates the questions, on the number of firms in each oligopoly and on the type of interaction which is assumed between firms that are integrated and firms that are not. In particular, it is shown that if no restriction is put on trade between integrated and non integrated firms, integrated firms may continue to purchase inputs from the non integrated upstream firms, with the goal of raising their downstream rival's costs. Furthermore, even though firms are identical, asymmetric equilibria, where integrated and non integrated firms coexist, may actually arise as an outcome of the integration game.